NYU's Hartry Field interviewed...

The role of deductive logic in rationality is often exaggerated. The notion of rationality isn’t very precise (and indeed doesn’t have fixed descriptive content: ‘rational’ is an approval-term), but logic is one of only many factors. For instance there could probably be a perfectly logical advocate of the claim that the moon landing was a hoax, but few would regard such a person rational. It’s frequently noted that when one becomes aware that certain of one’s beliefs are logically inconsistent, the most rational response is often to keep the inconsistency, because one is not sure how best to eliminate it. The ability to manage known inconsistencies and other tensions in one’s beliefs is one of the many important factors in rationality that stress on “being logical” obscures.